2. Consider the sets A = {1,2,3,4} and B = {1,3}. And consider the relation k...
2. Consider the sets A = {1,2,3,4} and B = {1,3}. And consider the relation k on the power set of A, P(A). Where: VX,Y E P(A) XkYXnB = Y OB Show that k is an equivalence relation on the power set of A.
8. On the set A = {1,2,3,4,...,20}, an equivalence relation R is defined as follows: For all x, y € A, xRy 4(x - y). For each of the following, circle TRUE or FALSE. [4 points) a. TRUE or FALSE: There are only 4 distinct equivalence classes for this relation. b. TRUE or FALSE: If you remove all the even numbers from A, the relation would still be an equivalence relation. C. TRUE or FALSE: In this equivalence relation, 2R5...
2. Consider the relation E on Z defined by E n, m) n+ m is even} equivalence relation (a) Prove that E is an (b) Let n E Z. Find [n]. equivalence relation in [N, the equivalence class of 3. We defined a relation on sets A B. Prove that this relation is an (In this view, countable sets the natural numbers under this equivalence relation). exactly those that are are 2. Consider the relation E on Z defined by...
List the members of the equivalence relation on {1,2,3,4}. Find the equivalence classes [1],[2],[3],[4] for the followi {{1},{2},{3},{4}} Determine whether each relation is reflexive,antisymmetric , or transitive (x,y) in R if xy>1 (x,y) in R if x > y (x,y) in R if 3 divides x + 2y
Consider the following definition of equivalent sets of functional dependencies on a relation: “Two sets of functional dependencies F and F’ on a relation R are equivalent if all FD’s in F’ follow from the ones in F, and all the FD’s in F follow from the ones in F’.” Given a relation R(A, B, C) with the following sets of functional dependencies: F1 = {A B, B C}, F2 = {A B, A C}, and...
Question 8 (1 point) Which of the following sets is equal to {1,3}? o {2x + 1 € Z1-3 < x² < 3} {x² + 1 € Z|XE Z and x < 2} . o {x E N | 2x + 1 <2} o {x? EN|0 < 2x +1 < 3} O None of the above Question 7 (1 point) Let Α = {a, b, c} and consider the equivalence relation R = {(a, 1), (b, b), (c, c), (c,...
Let us consider 2 sets A = {1,2,3,4} and B = { 5,6}. 1. What is {}? 2. What is |A|? 3. What is A union B? 4. What is A intersection B? 5. What is {{}}? 6. What is |{{}}|? 7. What is A x B? 8. What is BxA? 9. If A has 2^5 elements and B has 2^6 elements, how many elements are there in AxB?
Consider the following relation R on the set A = {1,2,3,4,5}. R= {(1, 1), (2, 2), (2, 3), (3, 2), (3, 3), (4,4), (4,5), (5,4), (5,5)} Given that R is an equivalence relation on A, which of the following is the partition of A into equivalence classes? Select the correct response. A. P = {{1}, {1, 2}, {3}, {3,4}, {4},{5}} B. P ={{1,2,3,4,5}} C. P ={{1,2},{3,4}, {5}} D. P = {{1}, {2,3}, {4,5}} E. P ={{1,2,3}, {1,5}} F. P= {{1},...
17-26 true or false questions 17. The smallest positive real number is c, where c = card(0,1). 18. To show that two sets A and B are equal, show that x A and x B. 19. If (vx)P(e) is false, then P(x) is never true for that domain. 20. If R is a relation on A and if (a, a) is true for some a in A, then R is reflexive. 21. If f:A → B is a function, then...
Let A={1,2,3,4}. Pick a subset B⊆A uniformly among the 2^4 subsets (i.e. the power set ofA) and let X be its size. Then likewise pick a subset C⊆B uniformly from the power set of B and let Y be its size. Give the joint p.m.f of (X, Y) and compute E(X−Y). Hint: X, Y can take value 0 if you pick the empty set. You can either write down a table or a compact expression of the form P(X=i, Y=j).